Generally, an FFT analyzer extracts a segment of a continuous time waveform for a period of T seconds (where T is called the sampling time) and performs a finite Fourier transform. The extracted time waveform can then be considered a periodic function with period T and decomposed into its fundamental frequency 1/T (the reciprocal of the period) and its integer multiples of that frequency. (See Figure 1) This T-second sampling time becomes the time window length of the FFT.
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Figure 1. The spectrum of the fundamental frequency (1/T) and its integer multiples.
This fundamental frequency represents the ability to resolve the spectrum, and we will call it the resolution frequency, denoted by Δf.
Intuitively, Δf(Hz) represents the lowest frequency that can be analyzed. Even if a component has a frequency lower than Δf(Hz) (in other words, a period longer than T seconds), it will be outside the analysis range. For example, with a time window length of 2 seconds (T=2), the lowest frequency that can be analyzed is 0.5Hz. Therefore, even if a frequency component lower than 0.5Hz is present within a 2-second time window, it cannot be analyzed. The lower limit of the analyzable frequency (excluding DC components) is thus limited by the time window length.
Furthermore, the frequency spectrum is structured as a series of filters, each with a width of Δf, arranged at equal intervals. These filters are sometimes called FFT bins. The term "bin" in English refers to a bucket or container. In FFT, the bin width is Δf, and only the DC component has a width of Δf/2. (Figure 2)
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Figure 2. Arrangement of bins with width Δf.
Next, let's consider what the maximum frequency and number of bins we can analyze are.
Since the time window length T is, in reality, digital time data, and it is obtained by recording N points with a certain sampling period Δt (or sampling frequency fS), the time window length (sampling time) T is,
It can be expressed as follows: Here, N is the number of points calculated in the FFT. From equations (1) and (2)
From equation (3), if the sampling frequency and the number of FFT calculation points (or sampling points) N are determined, the resolution frequency Δf is uniquely determined.
The maximum frequency that can be analyzed is theoretically fS/2, according to the Nyquist sampling theorem. This frequency is sometimes called the Nyquist frequency. Similarly, the number of bins can be determined up to N/2. In actual FFT analyzers, the effective analysis range (fSPAN) is set to fS/2.56, which is smaller than the Nyquist frequency fS/2, as the range in which aliasing (see technical glossary) error can be ignored.
Glossary of basic terms related to aliasing-FFT analysis
In an FFT analyzer, this value is called the analysis frequency range.
If we let the number of bottles be L, then similarly
The value in this bin is commonly referred to as the number of analysis lines.
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Figure 3. Nyquist frequency and effective analysis range (fSPAN)
Furthermore, Δf is obtained from equations (4) and (5).
It can also be expressed as follows.
In actual FFT analyzers (our DS series), the user specifies the analysis frequency range and the number of sampling points.
As a specific numerical example,
Analysis frequency range: 10kHz
Number of samples: 2048
So,
From equation (4), fS = 25.6 (kHz)
From equation (5), L = 800
(Note) Including DC, L=801
From equation (1) or (6), Δf = 12.5 (Hz)
This is how you can calculate it.
Next, how can we increase the frequency resolution (i.e., reduce Δf)?
From equation (3), it is clear that we must either lower the sampling frequency (i.e., reduce the frequency range) or increase the number of sampling points. Lowering the frequency range presents the problem that high-frequency bands cannot be analyzed with high resolution. Frequency zooming is a solution to this problem.
Frequency zooming involves using a high frequency of interest as the center frequency, shifting it to a lower frequency using a digital heterodyne method, applying a digital filter, and then resampling (decimation) to effectively lower the sampling frequency for analysis. Even with this method, the original time waveform still requires a longer duration equal to the zoom magnification, and equation (1) holds true.
If we can assume that the analysis is of a periodic signal with a constant frequency spectrum (i.e., a line spectrum), then we can use a search-enhancement function that allows us to read the peak frequency of the line spectrum with increased resolution by interpolating the values before and after the peak of the line spectrum from the shape of the Hanning window.
(Excerpt from the email newsletter issued on May 22, 2008)
